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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2405.20292 |
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| _version_ | 1866909332047659008 |
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| author | He, Song Huang, Yu-tin Kuo, Chia-Kai |
| author_facet | He, Song Huang, Yu-tin Kuo, Chia-Kai |
| contents | In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an $\ell$-loop geometry is attached. The loop geometry then consists of $\ell$ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the $\textit{loop geometry}$ can be viewed as fibration over a $\textit{tree geometry}$. The fibration naturally dissects the base into chambers, in which the degree-$4 \ell$ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of $x^2_{1,2}x^2_{3,4}$, $x^2_{1,4}x^2_{2,3}$ and $x^2_{1,3}x^2_{2,4}$. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to $\ell=3$, which indeed yield correct loop integrands for the four-point correlator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | All-loop geometry for four-point correlation functions He, Song Huang, Yu-tin Kuo, Chia-Kai High Energy Physics - Theory In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an $\ell$-loop geometry is attached. The loop geometry then consists of $\ell$ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the $\textit{loop geometry}$ can be viewed as fibration over a $\textit{tree geometry}$. The fibration naturally dissects the base into chambers, in which the degree-$4 \ell$ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of $x^2_{1,2}x^2_{3,4}$, $x^2_{1,4}x^2_{2,3}$ and $x^2_{1,3}x^2_{2,4}$. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to $\ell=3$, which indeed yield correct loop integrands for the four-point correlator. |
| title | All-loop geometry for four-point correlation functions |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2405.20292 |